Cesàro summable difference sequence space

Abstract

The difference sequence spaces $c_0(\mathrm{\Delta })$, $c(\mathrm{\Delta })$ and ${\ell }_{\mathrm{\infty }}(\mathrm{\Delta })$ were introduced by Kizmaz (Can. Math. Bull. 24:169-176, 1981). In this paper, we introduce the Cesáro summable difference sequence space $C_1(\mathrm{\Delta })$ which strictly includes the spaces $c_0(\mathrm{\Delta })$ and $c(\mathrm{\Delta })$ but overlaps with ${\ell }_{\mathrm{\infty }}(\mathrm{\Delta })$. It is shown that the newly introduced space $C_1(\mathrm{\Delta })$ turns out to be an inseparable BK space which does not possess any of the following: AK property, monotonicity, normality and perfectness. The Köthe-Toeplitz duals of $C_1(\mathrm{\Delta })$ are computed and as an application, the matrix classes $(C_1(\mathrm{\Delta }),{\ell }_{\mathrm{\infty }})$, $(C_1(\mathrm{\Delta }),c;P)$ and $(C_1(\mathrm{\Delta }),c_0)$ are also characterized.

MSC:40C05, 40A05, 46A45.

1 Notations and definitions

By s we shall denote the linear space of all complex sequences over ℂ (the field of complex numbers). ${\ell }_{\mathrm{\infty }}$, c and $c_0$ denote the spaces of all bounded, convergent and null sequences $x=(x_k)$ with complex terms, respectively, normed by ${\parallel x\parallel }_{\mathrm{\infty }}={sup}_k|x_k|$.

Throughout this paper, unless otherwise specified, we write ${\sum }_k$ for ${\sum }_{k=1}^{\mathrm{\infty }}$ and ${lim}_n$ for ${lim}_{n\to \mathrm{\infty }}$.

The definitions given below may be conveniently found in [1–3].

A complete metric linear space is called a Frèchet space. Let X be a linear subspace of s such that X is a Frèchet space with continuous coordinate projections. Then we say that X is an FK space. If the metric of an FK space is given by a complete norm, then we say that X is a BK space.

We say that an FK space X has AK, or has the AK property, if $(e_k)$, the sequence of unit vectors, is a Schauder basis for X.

A sequence space X is called

• (i) normal (or solid) if $y=(y_k)\in X$ whenever $|y_k|\le |x_k|$, $k\ge 1$, for some $x=(x_k)\in X$,

• (ii) monotone if it contains the canonical preimages of all its stepspaces,

• (iii) sequence algebra if $xy=(x_k{y}_k)\in X$ whenever $x=(x_k),y=(y_k)\in X$,

• (iv) convergence free when, if $x=(x_k)$ is in X and if $y_k=0$ whenever $x_k=0$, then $y=(y_k)$ is in X.

The idea of dual sequence spaces was introduced by Köthe and Toeplitz [4] whose main results concerned α-duals; the α-dual of $X\subset s$ being defined as

$$X^{\alpha }=\{a=(a_k)\in s:\sum _k|a_k{x}_k|<\mathrm{\infty }\text{ for all }x=(x_k)\in X\}.$$

In the same paper [4], they also introduced another kind of dual, namely, the β-dual (see [5] also, where it is called the g-dual by Chillingworth) defined as

$$X^{\beta }=\{a=(a_k)\in s:\sum _k{a}_k{x}_k\text{ converges for all }x=(x_k)\in X\}.$$

Obviously, $\varphi \subset X^{\alpha }\subset X^{\beta }$, where ϕ is the well-known sequence space of finitely non-zero scalar sequences. Also, if $X\subset Y$, then $Y^{\eta }\subset X^{\eta }$ for $\eta =\alpha ,\beta$. For any sequence space X, we denote ${(X^{\delta })}^{\eta }$ by $X^{\delta \eta }$, where $\delta ,\eta =\alpha \text{ or }\beta$. It is clear that $X\subset X^{\eta \eta }$, where $\eta =\alpha \text{ or }\beta$.

For a sequence space X, if $X=X^{\alpha \alpha }$ then X is called a Köthe space or a perfect sequence space.

A sequence space $x=(x_k)$ of complex numbers is said to be $(C,1)$ summable (or Cesàro summable of order 1) to $\ell \in \mathbb{C}$ if ${lim}_k{\sigma }_k=\ell$, where ${\sigma }_k=\frac{1}{k}{\sum }_{i=1}^k{x}_i$. By $C_1$ we shall denote the linear space of all $(C,1)$ summable sequences of complex numbers over ℂ, i.e.,

$$C_1=\{x=(x_k)\in s:\left(\frac{1}{k}\sum _{i=1}^k{x}_i\right)\in c\}.$$

It is easy to see that $C_1$ is a BK space normed by

$$\parallel x\parallel =\underset{k}{sup}\frac{1}{k}\left|\sum _{i=1}^k{x}_i\right|,x=(x_k)\in C_1.$$

The notion of difference sequence space was introduced by Kizmaz [6] in 1981 as follows:

$$X(\mathrm{\Delta })=\{x=(x_k)\in s:(\mathrm{\Delta }{x}_k)\in X\}$$

for $X={\ell }_{\mathrm{\infty }},c,c_0$; where $\mathrm{\Delta }{x}_k=x_k-x_{k+1}$ for all $k\in \mathbb{N}$ (the set of natural numbers). For a detailed account of difference sequence spaces, one may refer to [7–18] where many more references can be found.

2 Motivation and introduction

During the last 32 years, a large amount of work has been carried out by many mathematicians regarding various generalizations of difference sequence spaces of Kizmaz. Keeping aside some exceptions (see, for instance, [7, 8]), in most of these works, the underlying spaces have remained the same, i.e., ${\ell }_{\mathrm{\infty }}$, c and $c_0$. In the present work, we take the opportunity to introduce a difference sequence space with underlying space as $C_1$.

We observe that

• (i) $C_1⊈c(\mathrm{\Delta })$ as $({(-1)}^k)\in C_1$ but $({(-1)}^k)\notin c(\mathrm{\Delta })$,

• (ii) $c(\mathrm{\Delta })⊈C_1$ as $(k)\in c(\mathrm{\Delta })$ but $(k)\notin C_1$, and

• (iii) $c\subset c(\mathrm{\Delta })\cap C_1$.

Thus the sequence spaces $C_1$ and $c(\mathrm{\Delta })$ overlap but do not contain each other. Similarly, $C_1$ and ${\ell }_{\mathrm{\infty }}$ also overlap without containing each other as is clear from the fact that $C_1⊈{\ell }_{\mathrm{\infty }}$, ${\ell }_{\mathrm{\infty }}⊈C_1$ and $c\subset C_1\cap {\ell }_{\mathrm{\infty }}$. Note that the sequence $({(-1)}^{k-1}\sqrt{k})$ is $(C,1)$ summable but not bounded, whereas the sequence $x=(x_k)$ given by $x_1=1$, $x_2=0$ and

$$x_k=\{\begin{array}{cc}1,\hfill & \text{if }{2}^{i-1}<k\le 3(2^{i-2})(i=2,3,\dots );\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}$$

is bounded but not $(C,1)$ summable. This has motivated the authors to look for a new sequence space which properly includes the spaces $C_1$, $c(\mathrm{\Delta })$ and ${\ell }_{\mathrm{\infty }}$.

We now introduce a sequence space $C_1(\mathrm{\Delta })$, Cesàro summable difference sequence space, as follows:

$$C_1(\mathrm{\Delta })=\{x=(x_k)\in s:(\mathrm{\Delta }{x}_k)\in C_1\}.$$

The overall picture regarding inclusions among the already existing spaces ${\ell }_{\mathrm{\infty }}$, c, $c_0$, $C_1$, ${\ell }_{\mathrm{\infty }}(\mathrm{\Delta })$, $c(\mathrm{\Delta })$, $c_0(\mathrm{\Delta })$ and the newly introduced space $C_1(\mathrm{\Delta })$ is as shown below:

$$\begin{array}{ccc}{C}_1& \subset & C_1(\mathrm{\Delta })\\ \cup & \cup \\ c_0& \subset & c& \subset & {\ell }_{\mathrm{\infty }}\\ \cap & \cap & \cap \\ c_0(\mathrm{\Delta })& \subset & c(\mathrm{\Delta })& \subset & {\ell }_{\mathrm{\infty }}(\mathrm{\Delta })\\ \cap \\ C_1(\mathrm{\Delta })\end{array}$$

In this paper we show that $C_1(\mathrm{\Delta })$ strictly includes the spaces $c_0(\mathrm{\Delta })$ and $c(\mathrm{\Delta })$ but overlaps with ${\ell }_{\mathrm{\infty }}(\mathrm{\Delta })$. It is shown that the newly introduced space $C_1(\mathrm{\Delta })$ is an inseparable BK space which does not possess any of the following: AK property, monotonicity, normality and perfectness. The Köthe-Toeplitz duals of $C_1(\mathrm{\Delta })$ are computed, and as an application, the matrix classes $(C_1(\mathrm{\Delta }),{\ell }_{\mathrm{\infty }})$, $(C_1(\mathrm{\Delta }),c;P)$ and $(C_1(\mathrm{\Delta }),c_0)$ are also characterized.

3 Inclusion theorems and topological properties of $C_1(\mathrm{\Delta })$

We begin with elementary inclusion theorems justifying that $C_1(\mathrm{\Delta })$ is much wider than ${\ell }_{\mathrm{\infty }}$, $C_1$ and $c(\mathrm{\Delta })$.

Theorem 3.1 ${\ell }_{\mathrm{\infty }}\subset C_1(\mathrm{\Delta })$, the inclusion being strict.

Proof Let $x=(x_k)\in {\ell }_{\mathrm{\infty }}$. Then there exists $M>0$ such that $|x_1-x_{k+1}|\le M$ for all $k\ge 1$, and so $\frac{1}{k}{\sum }_{i=1}^k\mathrm{\Delta }{x}_i\to 0$ as $k\to \mathrm{\infty }$. For strict inclusion, observe that $(k)\in C_1(\mathrm{\Delta })$ but $(k)\notin {\ell }_{\mathrm{\infty }}$. □

Theorem 3.2 $C_1\subset C_1(\mathrm{\Delta })$, the inclusion being strict.

Proof For $x=(x_k)\in C_1$, we have ${lim}_k\frac{1}{k}{x}_k=0$, and so $\frac{1}{k}{\sum }_{i=1}^k\mathrm{\Delta }{x}_i\to 0$ as $k\to \mathrm{\infty }$. Inclusion is strict in view of the example cited in Theorem 3.1. □

Theorem 3.3 $c(\mathrm{\Delta })\subset C_1(\mathrm{\Delta })$, the inclusion being strict.

Proof Inclusion is obvious since $c\subset C_1$. To see that the inclusion is strict, consider the sequence $x=(x_k)=(1,2,1,2,1,2,\dots )$. □

Remark 3.4 Let X and Y be sequence spaces. If $X⊈Y$, then $X(\mathrm{\Delta })⊈Y(\mathrm{\Delta })$.

Proof Since $X⊈Y$, there is a sequence $x=(x_k)\in X$ such that $x\notin Y$. Consider the sequence $y=(y_k)=(0,-x_1,-x_1-x_2,-x_1-x_2-x_3,\dots )$. Then $y\in X(\mathrm{\Delta })$ but $y\notin Y(\mathrm{\Delta })$. □

Remark 3.5 We have already observed that $C_1⊈{\ell }_{\mathrm{\infty }}$ and ${\ell }_{\mathrm{\infty }}⊈C_1$, so by Remark 3.4, it follows that neither $C_1(\mathrm{\Delta })\subseteq {\ell }_{\mathrm{\infty }}(\mathrm{\Delta })$ nor ${\ell }_{\mathrm{\infty }}(\mathrm{\Delta })\subseteq C_1(\mathrm{\Delta })$. Also, we have $c(\mathrm{\Delta })\subset C_1(\mathrm{\Delta })\cap {\ell }_{\mathrm{\infty }}(\mathrm{\Delta })$. In view of this and Theorem 3.3, we can say that $C_1(\mathrm{\Delta })$ strictly includes $c(\mathrm{\Delta })$ and hence $c_0(\mathrm{\Delta })$ but overlaps with ${\ell }_{\mathrm{\infty }}(\mathrm{\Delta })$.

We now study the linear topological structure of $C_1(\mathrm{\Delta })$.

Theorem 3.6 $C_1(\mathrm{\Delta })$ is a BK space normed by

$${\parallel x\parallel }_{\mathrm{\Delta }}=|x_1|+\underset{k}{sup}\frac{1}{k}\left|\sum _{i=1}^k\mathrm{\Delta }{x}_i\right|,x=(x_k)\in C_1(\mathrm{\Delta }).$$

The proof is a routine verification by using ‘standard’ techniques and hence is omitted.

Theorem 3.7 $C_1(\mathrm{\Delta })$ is not separable.

Proof Let A be the set of all sequences $x_a,x_b,\dots$ , where

$$x_a={(k+a)}_k=(1+a,2+a,\dots ),x_b={(k+b)}_k=(1+b,2+b,\dots ),\dots$$

with $|a-b|>\frac{1}{2}$; $a,b\in \mathbb{R}$. Clearly, $A\subset C_1(\mathrm{\Delta })$ and A is uncountable. Let D be any dense set in $C_1(\mathrm{\Delta })$.

Define a map $f:A\to D$ as follows:

Let $x_a\in A\subset C_1(\mathrm{\Delta })$. As D is dense in $C_1(\mathrm{\Delta })$, so there exists some $z_{x_a}\in D$ such that ${\parallel x_a-z_{x_a}\parallel }_{\mathrm{\Delta }}<\frac{1}{4}$.

We set $f(x_a)=z_{x_a}$.

For $x_a,x_b\in A$, we have

$$\begin{array}{rcl}{\parallel x_a-x_b\parallel }_{\mathrm{\Delta }}& =& |(1+a)-(1+b)|+\underset{k}{sup}\frac{1}{k}\left|\sum _{i=1}^k\mathrm{\Delta }{(x_a-x_b)}_i\right|\\ \ge & |a-b|\\ >& \frac{1}{2}.\end{array}$$

Now

$$\begin{array}{rcl}{\parallel z_{x_a}-x_b\parallel }_{\mathrm{\Delta }}& \ge & {\parallel x_a-x_b\parallel }_{\mathrm{\Delta }}-{\parallel x_a-z_{x_a}\parallel }_{\mathrm{\Delta }}\\ >& \frac{1}{2}-\frac{1}{4}=\frac{1}{4}\end{array}$$

and already we have ${\parallel x_b-z_{x_b}\parallel }_{\mathrm{\Delta }}<\frac{1}{4}$, therefore $z_{x_a}\ne z_{x_b}$. Hence f is one-to-one. As $f(A)\subset D$ so D is uncountable. Thus, $C_1(\mathrm{\Delta })$ has no countable dense set. □

Corollary 3.8 $C_1(\mathrm{\Delta })$ does not have a Schauder basis.

The result follows from the fact that if a normed space has a Schauder basis, then it is separable.

Corollary 3.9 $C_1(\mathrm{\Delta })$ does not have the AK property.

Theorem 3.10 $C_1(\mathrm{\Delta })$ is not normal (solid) and hence neither perfect nor convergence free.

Proof Taking $x=(x_k)=(k-1)$ and $y=(y_k)=({(-1)}^k(k-1))$, we see that $x\in C_1(\mathrm{\Delta })$ but $y\notin C_1(\mathrm{\Delta })$ although $|y_k|\le |x_k|$, $k\ge 1$ and so $C_1(\mathrm{\Delta })$ is not normal. It is well known [1] that every perfect space, and also every convergence free space, is normal and consequently $C_1(\mathrm{\Delta })$ is neither perfect nor convergence free. □

Theorem 3.11 $C_1(\mathrm{\Delta })$ is neither monotone nor a sequence algebra.

Proof Take $x=(x_k)=(k)\in C_1(\mathrm{\Delta })$. Consider $y=(y_k)$ where

$$y_k=\{\begin{array}{cc}{x}_k,\hfill & \text{if }k\text{ is odd};\hfill \\ 0,\hfill & \text{if }k\text{ is even}\hfill \end{array}$$

i.e., $y=(1,0,3,0,5,\dots )$. Then $(\mathrm{\Delta }{y}_k)=(1,-3,3,-5,5,\dots )$ and so $(\mathrm{\Delta }{y}_k)\notin C_1$, i.e., $(y_k)\notin C_1(\mathrm{\Delta })$ and hence $C_1(\mathrm{\Delta })$ is not monotone. To see that $C_1(\mathrm{\Delta })$ is not a sequence algebra, take $x=y=(k)$ and observe that $x,y\in C_1(\mathrm{\Delta })$ but $xy=(k^2)\notin C_1(\mathrm{\Delta })$. □

4 Köthe-Toeplitz duals of $C_1(\mathrm{\Delta })$

In this section we compute the Köthe-Toeplitz duals of $C_1(\mathrm{\Delta })$ and show that $C_1(\mathrm{\Delta })$ is not perfect.

Theorem 4.1

$${[C_1(\mathrm{\Delta })]}^{\alpha }=\{a=(a_k):\sum _{k}k|a_k|<\mathrm{\infty }\}=D_1.$$

Proof Let $a=(a_k)\in D_1$. For any $x=(x_k)\in C_1(\mathrm{\Delta })$, we have $(\frac{1}{k}{\sum }_{i=1}^k\mathrm{\Delta }{x}_i)\in c$, i.e., $(\frac{1}{k}(x_1-x_{k+1}))\in c$ and so there exists some $M>0$ such that $|x_k|\le M(k-1)+x_1$ for $k\ge 1$ and hence ${sup}_k{k}^{-1}|x_k|<\mathrm{\infty }$, which implies that

$$\sum _k|a_k{x}_k|=\sum _k(k|a_k|)(k^{-1}|x_k|)<\mathrm{\infty }.$$

Thus, $a=(a_k)\in {[C_1(\mathrm{\Delta })]}^{\alpha }$.

Conversely, let $a=(a_k)\in {[C_1(\mathrm{\Delta })]}^{\alpha }$. Then ${\sum }_k|a_k{x}_k|<\mathrm{\infty }$ for all $x=(x_k)\in C_1(\mathrm{\Delta })$. Taking $x_k=k$ for all $k\ge 1$, we have $x=(x_k)\in C_1(\mathrm{\Delta })$ whence ${\sum }_{k}k|a_k|<\mathrm{\infty }$. □

Remark 4.2 It is well known [6, 16] that ${[c_0(\mathrm{\Delta })]}^{\alpha }={[c(\mathrm{\Delta })]}^{\alpha }={[{\ell }_{\mathrm{\infty }}(\mathrm{\Delta })]}^{\alpha }=D_1$, so we conclude that ${[c_0(\mathrm{\Delta })]}^{\alpha }={[c(\mathrm{\Delta })]}^{\alpha }={[{\ell }_{\mathrm{\infty }}(\mathrm{\Delta })]}^{\alpha }={[C_1(\mathrm{\Delta })]}^{\alpha }$, i.e., the α-duals of $c_0(\mathrm{\Delta })$, $c(\mathrm{\Delta })$, ${\ell }_{\mathrm{\infty }}(\mathrm{\Delta })$ and $C_1(\mathrm{\Delta })$ coincide.

Theorem 4.3

$${[C_1(\mathrm{\Delta })]}^{\alpha \alpha }=\{a=(a_k):\underset{k}{sup}{k}^{-1}|a_k|<\mathrm{\infty }\}=D_2.$$

Proof Taking $m=1$ and $X=c$ in [[12], Theorem 2.13], we have ${[c(\mathrm{\Delta })]}^{\alpha \alpha }=\{a=(a_k):{sup}_k{k}^{-1}|a_k|<\mathrm{\infty }\}$ and the result follows in view of Remark 4.2. □

Corollary 4.4 $C_1(\mathrm{\Delta })$ is not perfect.

The proof follows at once when we observe that the sequence $({(-1)}^k(k-1))\in {[C_1(\mathrm{\Delta })]}^{\alpha \alpha }$ but does not belong to $C_1(\mathrm{\Delta })$.

Theorem 4.5

$${[C_1(\mathrm{\Delta })]}^{\beta }=\{a=(a_k):\sum _{k}k|a_k|<\mathrm{\infty }\}=D_3.$$

Proof Let $a=(a_k)\in D_3$ and $x=(x_k)\in C_1(\mathrm{\Delta })$. Then $(\frac{1}{k}{\sum }_{i=1}^k\mathrm{\Delta }{x}_i)\in c$. For $n\in \mathbb{N}$, we have

$$\sum _{k=1}^n{a}_k{x}_k=-\sum _{k=2}^n(k-1)a_k\left(\frac{1}{k-1}\sum _{i=1}^{k-1}\mathrm{\Delta }{x}_i\right)+x_1\sum _{k=1}^n{a}_k.$$

Obviously, $(a_k)$ and $((k-1)a_k)\in {\ell }_1$. We define $y=(y_k)$ by $y_1=0$ and $y_k=\frac{1}{k-1}{\sum }_{i=1}^{k-1}\mathrm{\Delta }{x}_i$ for all $k\ge 2$. Then $y\in c$ and since $c^{\alpha }={\ell }_1$, the series ${\sum }_{k=2}^{\mathrm{\infty }}(k-1)a_k(\frac{1}{k-1}{\sum }_{i=1}^{k-1}\mathrm{\Delta }{x}_i)$ converges absolutely.

Conversely, if $a=(a_k)\in {[C_1(\mathrm{\Delta })]}^{\beta }$, then ${\sum }_k{a}_k{x}_k$ converges for all $x=(x_k)\in C_1(\mathrm{\Delta })$. In particular, taking $x_k=1$ for all k, we have ${\sum }_k{a}_k$ converges and so ${\sum }_{k=2}^{\mathrm{\infty }}(k-1)a_k(\frac{1}{k-1}{\sum }_{i=1}^{k-1}\mathrm{\Delta }{x}_i)$ converges for all $x=(x_k)\in C_1(\mathrm{\Delta })$. Since $x=(x_k)\in C_1(\mathrm{\Delta })$ if and only if $y=(\frac{1}{k}{\sum }_{i=1}^k\mathrm{\Delta }{x}_i)\in c$, we have $((k-1)a_k)\in c^{\alpha }$. □

Corollary 4.6 ${[c_0(\mathrm{\Delta })]}^{\alpha }={[c(\mathrm{\Delta })]}^{\alpha }={[{\ell }_{\mathrm{\infty }}(\mathrm{\Delta })]}^{\alpha }={[C_1(\mathrm{\Delta })]}^{\alpha }={[C_1(\mathrm{\Delta })]}^{\beta }$.

5 Matrix maps

Finally, we characterize certain matrix classes. For any complex infinite matrix $A=(a_{nk})$, we shall write $A_n={(a_{nk})}_{k\in \mathbb{N}}$ for the sequence in the n th row of A. If X, Y are any two sets of sequences, we denote by $(X,Y)$ the class of all those infinite matrices $A=(a_{nk})$ such that the series $A_n(x)={\sum }_k{a}_{nk}{x}_k$ converges for all $x=(x_k)\in X$ ($n=1,2,\dots$) and the sequence $Ax={(A_{n}x)}_{n\in \mathbb{N}}$ is in Y for all $x\in X$.

The following theorem is well known.

Theorem 5.1 [[3], p.219] Let X and Y be BK spaces and suppose that $A=(a_{nk})$ is an infinite matrix such that ${({\sum }_k{a}_{nk}{x}_k)}_{n\in \mathbb{N}}\in Y$ for each $x\in X$, i.e., $A\in (X,Y)$, then $A:X\to Y$ is a bounded linear operator.

Theorem 5.2 $A\in (C_1(\mathrm{\Delta }),{\ell }_{\mathrm{\infty }})$ if and only if ${sup}_n{\sum }_{k=2}^{\mathrm{\infty }}(k-1)|a_{nk}|<\mathrm{\infty }$.

Proof Suppose that ${sup}_n{\sum }_{k=2}^{\mathrm{\infty }}(k-1)|a_{nk}|<\mathrm{\infty }$ and $x=(x_k)\in C_1(\mathrm{\Delta })$. Proceeding as in Theorem 4.5, we have ${\sum }_{k=2}^{\mathrm{\infty }}|a_{nk}{\sum }_{i=1}^{k-1}\mathrm{\Delta }{x}_i|<\mathrm{\infty }$.

For $m\in \mathbb{N}$,

$$\sum _{k=1}^m{a}_{nk}{x}_k=-\sum _{k=1}^m{a}_{nk}\left(\sum _{i=1}^{k-1}\mathrm{\Delta }{x}_i\right)+x_1\sum _{k=1}^m{a}_{nk},$$

which yields the absolute convergence of ${\sum }_k{a}_{nk}{x}_k$ for each $n\in \mathbb{N}$, and finally we have

$$\left|\sum _k{a}_{nk}{x}_k\right|\le \left(\underset{k\ge 2}{sup}\left|\frac{1}{k-1}\sum _{i=1}^{k-1}\mathrm{\Delta }{x}_i\right|\right)(\underset{n}{sup}\sum _{k=2}^{\mathrm{\infty }}(k-1)|a_{nk}|)+x_1\underset{n}{sup}\sum _k(k-1)|a_{nk}|$$

for all $n\in \mathbb{N}$.

Conversely, by Theorem 5.1, we have

$$\left|\sum _k{a}_{nk}{x}_k\right|=|A_n(x)|\le \underset{n}{sup}|A_n(x)|={\parallel Ax\parallel }_{\mathrm{\infty }}\le \parallel A\parallel {\parallel x\parallel }_{\mathrm{\Delta }}$$

(5.1)

for each $n\in \mathbb{N}$ and $x=(x_k)\in C_1(\mathrm{\Delta })$.

Choose any $n\in \mathbb{N}$ and any $r\in \mathbb{N}$ and define

$$x_k=\{\begin{array}{cc}(k-1)sgn{a}_{nk},\hfill & \text{if }1<k\le r;\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

Then $x=(x_k)\in c\subset C_1(\mathrm{\Delta })$ with ${\parallel x\parallel }_{\mathrm{\Delta }}=1$. Inserting this value of $x=(x_k)$ in (5.1) , we have

$$\sum _{k=2}^r(k-1)|a_{nk}|\le \parallel A\parallel .$$

(5.2)

Letting $r\to \mathrm{\infty }$ and noting that (5.2) holds for every $n\in \mathbb{N}$, we are through. □

Remark 5.3 If $x=(x_k)\in C_1(\mathrm{\Delta })$, then there exists some $\ell \in \mathbb{C}$ such that ${lim}_k\frac{1}{k}{\sum }_{i=1}^k\mathrm{\Delta }{x}_i=\ell$. We shall call ℓ the $C_1(\mathrm{\Delta })$ limit of the sequence $(x_k)$ and by $(C_1(\mathrm{\Delta }),c;P)$ we shall denote that subset of $(C_1(\mathrm{\Delta }),c)$ for which $C_1(\mathrm{\Delta })$ limits are preserved.

Theorem 5.4 $A\in (C_1(\mathrm{\Delta }),c;P)$ if and only if

• (i) ${sup}_n{\sum }_{k=2}^{\mathrm{\infty }}(k-1)|a_{nk}|<\mathrm{\infty }$,

• (ii) ${lim}_n{\sum }_k(k-1)a_{nk}=-1$,

• (iii) ${lim}_n{a}_{nk}=0$ for each k,

• (iv) ${lim}_n{\sum }_k{a}_{nk}=0$.

Proof Let the conditions (i)-(iv) hold and suppose that $x=(x_k)\in C_1(\mathrm{\Delta })$ with ${lim}_k\frac{1}{k}{\sum }_{i=1}^k\mathrm{\Delta }{x}_i=\ell$. It is implicit in (i) that, for each $n\in \mathbb{N}$, ${\sum }_k(k-1)|a_{nk}|$ converges. It follows that ${\sum }_{k=2}^{\mathrm{\infty }}(k-1)a_{nk}(\frac{1}{k-1}{\sum }_{i=1}^{k-1}\mathrm{\Delta }{x}_i)$ converges, whence

$$\sum _k{a}_{nk}{x}_k=-\sum _{k=2}^{\mathrm{\infty }}(k-1)a_{nk}(\frac{1}{k-1}\sum _{i=1}^{k-1}\mathrm{\Delta }{x}_i-\ell )-\ell \sum _k(k-1)a_{nk}+x_1\sum _k{a}_{nk}.$$

(5.3)

Let ${ϵ}_k=\frac{1}{k}{\sum }_{i=1}^k\mathrm{\Delta }{x}_i-\ell$, $H={sup}_k|{ϵ}_k|$ and $M={sup}_n{\sum }_k(k-1)|a_{nk}|$.

Then, for any $p\in \mathbb{N}$, we have

$$|\sum _{k=2}^{\mathrm{\infty }}(k-1)a_{nk}(\frac{1}{k-1}\sum _{i=1}^{k-1}\mathrm{\Delta }{x}_i-\ell )|\le H\sum _{k=2}^p(k-1)|a_{nk}|+M\underset{k>p}{sup}|{ϵ}_{k-1}|$$

and hence

$$\underset{n}{lim\hspace{0.17em}sup}|\sum _{k=2}^{\mathrm{\infty }}(k-1)a_{nk}(\frac{1}{k-1}\sum _{i=1}^{k-1}\mathrm{\Delta }{x}_i-\ell )|\le M\underset{k>p}{sup}|{ϵ}_{k-1}|.$$

Letting $p\to \mathrm{\infty }$, we have ${\sum }_{k=2}^{\mathrm{\infty }}(k-1)a_{nk}(\frac{1}{k-1}{\sum }_{i=1}^{k-1}\mathrm{\Delta }{x}_i-\ell )\to 0$ as $n\to \mathrm{\infty }$. Making use of this and also of (ii) and (iv) in (5.3), we get the result.

Conversely, let $A\in (C_1(\mathrm{\Delta }),c;P)$. Then ${({\sum }_k{a}_{nk}{x}_k)}_{n\in \mathbb{N}}\in c$ for all $x=(x_k)\in C_1(\mathrm{\Delta })$. By the same argument as in Theorem 5.2, we have ${sup}_n{\sum }_{k=2}^{\mathrm{\infty }}(k-1)|a_{nk}|<\mathrm{\infty }$. Taking $x=e_k\in C_1(\mathrm{\Delta })$, we get ${(a_{nk})}_{n\in \mathbb{N}}\in c$ with ${lim}_n{a}_{nk}=0$ for each k. Also, for $x=(k-1)$, we have ${({\sum }_k(k-1)a_{nk})}_{n\in \mathbb{N}}\in c$ with ${lim}_n{\sum }_k(k-1)a_{nk}=-1$, and finally $x=(1,1,1,\dots )\in C_1(\mathrm{\Delta })$ yields ${lim}_n{\sum }_k{a}_{nk}=0$. □

Theorem 5.5 $A\in (C_1(\mathrm{\Delta }),c_0)$ if and only if

• (i) ${sup}_n{\sum }_{k=2}^{\mathrm{\infty }}(k-1)|a_{nk}|<\mathrm{\infty }$,

• (ii) ${lim}_n{\sum }_k(k-1)a_{nk}=0$,

• (iii) ${lim}_n{a}_{nk}=0$ for each k,

• (iv) ${lim}_n{\sum }_k{a}_{nk}=0$.